Abstract
In this paper, we show that there exist families of curves (defined over an alge- braically closed field k of characteristic p> 2) whose Jacobians have interesting p-torsion. For example, for every 0 ≤ f ≤ g, we find the dimension of the locus of hyperelliptic curves of genus g with p-rank at most f . We also produce families of curves so that the p-torsion of the Jacobian of each fibre contains multiple copies of the group scheme αp. The method is to study curves which admit an action by (Z/2) n so that the quotient is a projective line. As a result, some of these families intersect the hyperelliptic locus Hg.
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